The q-rung orthopair fuzzy set is a powerful tool for depicting fuzziness and uncertainty, as compared to the Pythagorean fuzzy model. The aim of this paper is to present q-rung orthopair fuzzy competition graphs (q-ROFCGs) and their generalizations, including q-rung orthopair fuzzy k-competition graphs, p-competition q-rung orthopair fuzzy graphs and m-step q-rung orthopair fuzzy competition graphs with several important properties. The study proposes the novel concepts of q-rung orthopair fuzzy cliques and triangulated q-rung orthopair fuzzy graphs with real-life characterizations. In particular, the present work evolves the notion of competition number and m-step competition number of q-rung picture fuzzy graphs with algorithms and explores their bounds in connection with the size of the smallest q-rung orthopair fuzzy edge clique cover. In addition, an application is illustrated in the soil ecosystem with an algorithm to highlight the contributions of this research article in practical applications.

The purpose of this research study is to present some new operations, including rejection, symmetric difference, residue product, and maximal product of Pythagorean fuzzy graphs (PFGs), and to explore some of their properties. This research article introduces certain notions, including intuitionistic fuzzy graphs of 3-type (IFGs3T), intuitionistic fuzzy graphs of 4-type (IFGs4T), and intuitionistic fuzzy graphs of n-type (IFGsnT), and proves that every IFG(n − 1)T is an IFGnT (for $n\ge$ 2). Moreover, this study discusses the application of Pythagorean fuzzy graphs in decision making.

Picture fuzzy model is a generalized structure of intuitionistic fuzzy model in the sense that it not only assigns the membership and nonmembership values in the form of orthopair $(\mu ,\nu )$ to an element, but it assigns a triplet $(\mu ,\eta ,\nu ),$ where $\eta$ denotes the neutral degree and the difference $\pi =1-(\mu +\eta +\nu )$ indicates the degree of refusal. The q-rung picture fuzzy set( $\mathrm{q}$ -RPFS) provides a wide formal mathematical sketch in which uncertain and vague conceptual phenomenon can be precisely and rigorously studied because of its distinctive quality of vast representation space of acceptable triplets. This paper discusses some properties including edge regularity, total edge regularity and perfect edge regularity of q-rung picture fuzzy graphs (q-RPFGs). The work introduces and investigates these properties for square q-RPFGs and q-RPF line graphs. Furthermore, this study characterizes how regularity and edge regularity of q-RPFGs structurally relate. In addition, it presents the concept of ego-networks to extract knowledge from large social networks under q-rung picture fuzzy environment with algorithm.